Abstract

Th. M. Rassias (1984) proved that the norm defined over a real vector space is induced by an inner product if and only if for a fixed integer holds for all The aim of this paper is to extend the applications of the fixed point alternative method to provide a fuzzy stability for the functional equation which is said to be a functional equation associated with inner product spaces.

1. Introduction

Studies on fuzzy normed linear spaces are relatively recent in the field of fuzzy functional analysis. In 1984, Katsaras [1] first introduced the notion of fuzzy norm on a linear space and at the same year Wu and Fang [2] also introduced a notion of fuzzy normed space and gave the generalization of the Kolmogoroff normalized theorem for a fuzzy topological linear space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [3–6].

Nowadays, fixed point and operator theory play an important role in different areas of mathematics, and its applications, particularly in mathematics, physics, differential equation, game theory and dynamic programming. Since fuzzy mathematics and fuzzy physics along with the classical ones are constantly developing, the fuzzy type of the fixed point and operator theory can also play an important role in the new fuzzy area and fuzzy mathematical physics. Many authors [4, 7–9] have also proved some different type of fixed point theorems in fuzzy (probabilistic) metric spaces and fuzzy normed linear spaces. In 2003, Bag and Samanta [10] modified the definition of Cheng and Mordeson [11] by removing a regular condition. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed linear spaces [12].

One of the most interesting questions in the theory of functional analysis concerning the Ulam stability problem of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to an exact solution of the given functional equation?

The first stability problem concerning group homomorphisms was raised by Ulam [13] in 1940 and affirmatively solved by Hyers [14]. The result of Hyers was generalized by Aoki [15] for approximate additive function and by Th. M. Rassias [16] for approximate linear functions by allowing the difference Cauchy equation to be controlled by . Taking into consideration a lot of influence of Ulam, Hyers and Th. M. Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Th. M. Rassias is called the generalized Hyers-Ulam stability. In 1994, a generalization of Th. M. Rassias theorem was obtained by Găvruţa [17], who replaced by a general control function .

On the other hand, J. M. Rassias [18–25] considered the Cauchy difference controlled by a product of different powers of norm. However, there was a singular case; for this singularity a counterexample was given by Găvruţa [26]. This stability phenomen on is called the Ulam-Găvruţa-Rassias stability (see also [27]).

Theorem 1.1 (J. M. Rassias [18]). Let be a real normed linear space and a real complete normed linear space. Assume that is an approximately additive mapping for which there exist constants and such that and satisfies inequality
for all , then there exists a unique additive mapping satisfying
for all . If, in addition, is a mapping such that the transformation is continuous in for each fixed , then is an -linear mapping.

Very recently, K. Ravi [28] in the inequality (1.1) replaced the bound by a mixed one involving the product and sum of powers of norms, that is, .

For more details about the results concerning such problems and mixed product-sum stability (J. M. Rassias Stability), the reader is referred to [29–41].

Quadratic functional equations were used to characterize inner product spaces [42–45]. A square norm on an inner product space satisfies the important parallelogram equality
The functional equation
is related to a symmetric biadditive function [46, 47]. It is natural that this equation is called a quadratic functional equation, and every solution of the quadratic equation (1.4) is said to be a quadratic function.

It was shown by Th. M. Rassias [48] that the norm defined over a real vector space is induced by an inner product if and only if for a fixed integer as follows:
for all In [49], Park proved the generalized Hyers-Ulam stability of a functional equation associated with inner product spaces:
in fuzzy normed spaces.

The main objective of this paper is to prove the the generalized Hyers-Ulam stability of the following functional equation associated with inner product spaces
in fuzzy normed spaces, based on the fixed point method. Interesting new results concerning functional equations associated with inner product spaces have recently been obtained by Park et al. [50–52] and Najati and Th. M. Rassias [53] as well as for the fuzzy stability of a functional equation associated with inner product spaces by Park [49].

The stability of different functional equations in fuzzy normed spaces and random normed spaces has been studied in [20, 21, 54–77]. In this paper, we prove the generalized fuzzy stability of a functional equation associated with inner product spaces (1.7).

2. Preliminaries

We start our work with the following notion of fixed point theory. For the proof, refer to [78]. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. [79].

Let be a generalized metric space. An operator satisfies a Lipschitz condition with Lipschitz constant if there exists a constant such that for all If the Lipschitz constant is less than , then the operator is called a strictly contractive operator. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity.

We recall the following theorem by Margolis and Diaz.

Theorem 2.1. Suppose that one is given a complete generalized metric space and a strictly contractive function with Lipschitz constant , then for each given , either
or other exists a natural number such that(i) for all (ii)the sequence is convergent to a fixed point of ;(iii) is the unique fixed point of in (iv) for all

Next, we define the notion of a fuzzy normed linear space.

Let be a real linear space. A function is said to be a fuzzy norm on [10] if and only if the following conditions are satisfied: for all and if and only if for all if ; for all and all is a nondecreasing function on and for all

In the following we will suppose that is left continuous for every

A fuzzy normed linear space is a pair , where is a real linear space and is a fuzzy norm on

Let be a normed linear space, then
is a fuzzy norm on

Let be a fuzzy normed linear space. A sequence in is said to be convergent if there exists such that for all In that case, is called the limit of the sequence and we write .

A sequence in is called Cauchy if for each and each there exists such that . If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space.

From now on, let be a linear space, be a fuzzy normed space and be a fuzzy Banach space. For convenience, we use the following abbreviation for a given function :
for all , where is a fixed integer.

3. Fuzzy Approximation

In the following theorem, we prove the fuzzy stability of the functional equation (1.7) via fixed point method, for an even case.

Theorem 3.1. Let be a function such that, for some real number with . Suppose that an even function with satisfies the inequality
for all and all then there exists a unique quadratic function such that and
for all and all , where

Proof. Letting , , and in (3.1) and using the evenness of , we obtain
for all and all Interchanging with in (3.4) and using the evenness of , we obtain
for all and all It follows from (3.4) and (3.5) that
for all and all Setting , and in (3.1) and using the evenness of , we obtain
for all and all So we obtain from (3.6) and (3.7) that
for all and all So
for all and all Putting , and in (3.1), we get
for all and all It follows from (3.9) and (3.10) that
for all and all Letting in (3.7) and replacing by in the obtained inequality, we get
for all and all It follows from (3.9), (3.10), (3.11) and (3.12) that
for all and all Applying (3.11) and (3.13), we obtain
for all and all Setting , and in (3.1), we obtain
for all and all It follows from (3.14) and (3.15) that
for all and all Therefore
for all and all which implies that
for all and all . Let be the set of all even functions with and introduce a generalized metric on as follows:
where, as usual, . It is easy to show that is a generalized complete metric space [80].Without loss of generality, we consider . Let us now consider the function defined by for all and Let such that , then
that is, if we have This means that for all that is, is a strictly contractive self-function on with the Lipschitz constant It follows from (3.18) that
for all and all which implies that .Due to Theorem 2.1, there exists a function such that is a fixed point of , that is, for all Also, as implies the equality for all Setting and in (3.1), we obtain that
for all and all By letting in (3.22), we find that for all which implies = 0. Thus satisfies (1.7). Hence the function is quadratic (See Lemma of [53]).According to the fixed point alterative, since is the unique fixed point of in the set , is the unique function such that
for all and all Again using the fixed point alterative, we get
which implies the inequality
for all and all So
for all and all This completes the proof.

In the following theorem, we prove the fuzzy stability of the functional equation (1.7) via fixed point method, for an odd case.

Theorem 3.2. Let be a function such that for some real number with . Suppose that an odd function satisfies the inequality (3.1) for all and all then there exists a unique additive function such that and
for all and all , where

Proof. Letting , and in (3.1) and using the oddness of , we obtain that
for all and all Interchanging with in (3.29) and using the oddness of , we get
for all and all It follows from (3.29) and (3.30) that
for all and all Setting , , and in (3.1) and using the oddness of , we get
for all and all So we obtain from (3.31) and (3.32) that
for all and all Putting , and in (3.1), we obtain
for all and all It follows from (3.33) and (3.34) that
for all and all Replacing and by and in (3.35), respectively, we obtain
for all and all Therefore
for all and all which implies that
for all and all . Let be the set of all odd functions and introduce a generalized metric on as follows:
where, as usual, . So is a generalized complete metric space. We consider the function defined by for all and Let such that , then
that is, if we have This means that for all that is, is a strictly contractive self-function on with the Lipschitz constant It follows from (3.38) that
for all and all which implies that .Due to Theorem 2.1, there exists a function such that is a fixed point of , that is, for all Also, as implies the equality for all Setting and in (3.1), we obtain that
for all and all By letting in (3.42), we find that for all which implies . Thus satisfies (1.7). Hence the function is additive (see Lemma of [53]).The rest of the proof is similar to the proof of Theorem 3.1.

The main result of the paper is the following.

Theorem 3.3. Let be a function such that, for some real number with . Suppose that a function with satisfies (3.1) for all and all then there exist a unique quadratic function and a unique additive function such that
for all and all , where and are defined as in Theorems 3.1 and 3.2.

Proof. Let for all then
for all and Hence, in view of Theorem 3.1, there exists a unique quadratic function such that
for all and On the other hand, let for all then, by using the above method from Theorem 3.2, there exists a unique additive function such that
for all and Hence, (3.43) follows from (3.45) and (3.46).

Acknowledgment

The second author would like to thank the Office of Gifted Students at Semnan University for its financial support.